Usually, in addition to the phenomenological description through the Landau-Lifschitz-Gilbert (LLG) equation
[1]}, [2]}, [3]}, [4]}, [5]}, [6]},
STT and SP processes have been investigated by adopting Green functions and bosonic representations to describe the magnetic
material [7]}, [8]}, [9]}, [10]}.
The Holstein-Primakoff representation [11]}, [12]} allows representing the
spin operators as first-order creation/annihilation operators only in the magnon low-occupation limit. In the order hand,
if magnon interactions are relevant, higher-order terms should be considered,
which introduce complications in the development. In the SU(N) Schwinger
[13]}, [14]}, [12]} representation, each spin component is represented by pair combinations involving N flavors of
bosonic operators, which results in a four-order Hamiltonian. The quartic-order Hamiltonian terms are then decoupled by
introducing auxiliary fields (a mean-field approximation) whose values are determined by solving coupled self-consistent equations.
The Schwinger formalism provides good results in both ordered and disordered phases; however, in frustrated models, including
Gaussian fluctuations becomes necessary [16]}, [17]}, [18]}, [19]}. In addition, extra care
is required, mainly for three-dimensional models close to the transition temperature [20]}.
